Saturday, July 31, 2010

Quantity Flows For Structural Dynamics

1.0 IntroductionThis post presents an example of a model of structural economic dynamics. I consider what quantity flows would arise for an economy in which agents make decisions in which the economy smoothly reproduces. The solution for this exercise turns out to be dynamically unstable in the special case I use for illustration. I think this means that, if I solve this special case in a future post for one way of setting out the price system, the solution for prices will be stable. The model presented in this post illustrates the difficult discovery problems that are solved in successful economies.

2.0 TechnologyThis economy consists of two sectors. In the first sector, labor produces means of production with existing means of production. In the second sector, labor produces means of consumption with existing means of production. (I use steel as as a synecdoche for means of production and corn for means of consumption.) The technique in use in both sectors exhibits Constant Returns to Scale (CRS). Only circulating capital is modeled; the means of production are entirely consumed in producing the output. Table 1 shows the coefficients of production for the technique in use during the t-th year.

Table 1: The Technology

SteelIndustry

CornIndustry

Labor

a0,1(t) person-years

a0,2(t) person-years

Steel

a1,1(t) tons

a1,2(t) tons

Outputs

1 ton steel

1 bushel corn

The technique improves each year. That is, each coefficient of production decreases at a constant rate of 100 ci,j percent per year:

[ai,j(t) - ai,j(t + 1)]/ai,j(t) = ci,j

The above difference equation can be solved in closed form. The coefficients of production evolve as:

ai,j(t) = ai,j(0) (1 - ci,j)t

A more complex formulation might have non-constant percentage rates of decrease in the coeffients of production. For example, the percentage rate of decrease might be larger if the level of output of an industry was larger. Then one would be modeling "learning by doing" or endogenous growth, following in the tradition of Nicholas Kaldor and Kenneth Arrow. (Mainstream economists would cite Paul Romer's confusedbalderdash.)

3.0 Conditions for Smooth ReproductionLet q1(t) and q2(t) be the tons of steel and the bushels of corn, respectively, produced as output and available at the end of the t-th year. I want to consider the case in which the labor force is always fully employed, the proportions in which output is produced always turns out to be appropriate, and no excess capacity is ever created.

The gross output of corn each year is divided up between the workers and the capitalists and then consumed. The gross outputs of steel and corn in a given year determine, along with the coefficients of production, how much steel should have been produced in the previous year:

q1(t - 1) = a1,1(t) q1(t) + a1,2(t) q2(t)

The amount of labor employed in the t-th year is:

L(t) = a0,1(t) q1(t) + a0,2(t) q2(t),

where L(t) is the person-years of labor employed. In a general formulation, one might model the number of workers growing each year, but with increased productivity being taken partly in the form of decreased working hours per worker. For simplicity, I here model the labor force as a given constant:

L(t) = L*

The above equations specify a dynamic system. An initial condition needs to be specified for any solution path to be completely determined. I take the initial ratio of employment in the two sectors as a given parameter:

a0,1(0) q1(0)/a0,2(0) q2(0) = h

The model can be simplified by expressing one quantity flow in terms of other by use of the condition that labor is fully employed. Some algebraic manipulation yields a single difference equation for the output of steel:

q1(t) = [a1,2(t) L* - a0,2(t) q1(t - 1)]/d(t),

where

d(t) = [a0,1(t)a1,2(t) - a0,2(t)a1,1(t)]

If the coefficients of production were constant, the above would be a linear difference equation. If I recall my mathematics correctly, linear systems either blow up; decay to an equilibrium; or, for coefficients meeting an exact balance, generate a constant wave.

4.0 The Solution of a Special CaseI tried a numerical experiment to increase my understanding of this dynamical system. Accordingly, I chose some specific values for the model parameters. Table 2 gives the initial coefficients of production. The difference equation for gross steel outputs is simplified in that the coefficients of production in a sector decrease at the same constant rate. I chose the following rates of decrease:

c0,1 = c1,1 = 1/20

c0,2 = c1,2 = 1/40

Let the labor force be unity:

L* = 1

Finally, I carefully specified an initial condition:

a0,1(0) q1(0)/a0,2(0) q2(0) = 0.22335983

Table 2: The Initial Technology

SteelIndustry

CornIndustry

Labor

a0,1(0) = 1

a0,2(0) = 1

Steel

a1,1(0) = 1/10

a1,2(0) = 1/5

Outputs

1 ton steel

1 bushel corn

One can easily step through the first few years of the solution, thereby obtaining the start of a series for q1(t) and q2(t).The solution is dynamically unstable. I carefully chose the initial condition to get six years before the solution blows up. For the first five years, the output of steel grows over 3% and the output of corn grows over 14 1/2%, for a constant labor supply. This set of priorities is the reverse of what was typically achieved in no-longer actually existing socialism. When Imre Nagy, for example, tried to put Hungary on a new course, he was deposed. The distribution of labor, shown in Table 2, is not realistic for a developing capitalistic economy either. In practice, the labor force becomes steadily less concentrated in producing means of consumption and more in producing means of production. Still, I think, this model with a better choice of parameters and perhaps some generalizations can be quite interesting.